Aieee 2007 Math

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Career Launcher 8 – 12 / Solutions AIEEE Test Paper 2007 Page 1

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Mathematics

1. If (2, 3, 5) is one end of a diameter of the sphere 2 2 2x y z 6x 12y 2z 20 0,+ + − − − + = then the

coordinates of the other end of the diameter are(1) (4, 3, –3) (2) (4, 9, –3) (3) (4, –3, 3) (4) (4, 3, 5)

Sol. (2)

Centre of sphere is (3, 6, 1)Let other end is (x

1, y

1, z

1)

11

x 23 x 4

2

+∴ = ⇒ =

11y 3 6 y 9

2+ = ⇒ =

11

z 51 z – 3

2

+= ⇒ =

∴ Other end of diameter is (4, 9, – 3)

2. Let ( )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆa i j k, b i j 2k and c xi x 2 j k,= + + = − + = + − − . If the vector c lies in the plane of a and

b , then x equals

(1) – 2 (2) 0 (3) 1 (4) – 4

Sol. (1)

c, a and bH H H

3 are coplanar

c a b∴ = λ + µH H H

( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆxi x 2 j k i j k i j 2k+ − − = λ + + + µ − +

x⇒ = λ + µ ... (i)

x 2− = λ − µ ... (ii)

1 2− = λ + µ ... (iii)From (i) and (ii)λ = x – 1, µ = 1∴ From (iii) – 1 = x – 1 + 2⇒ x = – 2

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Method II

1 1 1

1 1 2 0

x x 2 1− =− −

( ) ( ) ( )1 1 2x 4 1 1 2x 1 x 2 x 0⇒ − + − − − + − + =

2x 4⇒ = −⇒ x = – 2

3. Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse.If the area of the triangle is 1, then the set of values which ‘k’ can take is given by(1) { – 3, – 2} (2) {1, 3} (3) {0, 2} (4) { – 1, 3}

Sol. (4)

∆ = 1

h k 11

1 1 1 12

2 1 1

⇒ =

( ) ( ) ( )h 1 1 k 1 2 1 1 2 2⇒ − − − + − = ±

⇒ k – 1 = ±2

⇒ k = 3 or – 1

4. Let P = ( – 1, 0), Q = (0, 0) and R = ( )3, 3 3 be three points. The equation of the bisector of the

angle PQR is

(1) x 3y 0+ = (2) 3x y 0+ = (3)3

x y 02

+ = (4)3

x y 02

+ =

Sol. (2)

X

Y

QP( – 1, 0)

P (3, 3 3)√

π / 3 π / 3

π /3

S

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Slope of QS, m = tan120°= 3−

y 3x= −

y 3x 0+ =

5. If one of the lines of ( )2 2 2my 1 m xy mx 0+ − − = is a bisector of the angle between the lines

xy = 0, then m is

(1) 2 2.1

2− (3) – 2 (4) 1

Sol. (4)

Joint equation of bisector of the lines xy = 0 is 2 2y x 0− =

Since ( )2 2 2my 1 m xy mx 0+ − − =

( )( )y mx my x 0⇒ − + =

⇒ One of the line is bisector of xy = 0⇒ m = 1

6. Let F(x) = f(x) +1

fx

, where ( )x

1

logtf x dt.

1 t=

+

∫ Then F(e) equals

(1) 2 (2)1

2(3) 0 (4) 1

Sol. (2)

1F(e) f(e) f

e

= +

e 1/ e

1 1

logt logtdt dt

1 t 1 t= +

+ +∫ ∫

1 2I I= +

1/ e

2

1

logtFor I dt

1 t=

+∫

Let2

1 1t dt dz

z z= ⇒ = −

When t = 1 ⇒ z = 1

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1t z e

e= ⇒ =

e

2 21

logz 1I . dz

1 z1z

− ∴ = −

+∫

( ) ( )

e e

1 1

logz logtdz dt

z 1 z t 1 t= =

+ +∫ ∫

1 2F(e) I I∴ = +

( )

e

1

logt logt dt1 t t 1 t = + + + ∫

e 1

1 0

logt 1dt s ds s logt; ds dt, when t 1. s 0 and t e. s 1

t t= = = = = = = =∫ ∫

12

0

s 1

2 2

= =

7. Let f : R IR→ be a function defined by f(x) = Min{x + 1, |x| + 1}. Then which of the following is

true?(1) f(x) is not differentiable at x = 0

(2) f(x) ≥ 1 for all x R∈(3) f(x) is not differentiable at x = 1.(4) f(x) is differentiable everywhere

Sol. (4)

X

Y

f(x) = Min{x + 1, |x| + 1}

y =

x + 1

O

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f(x) = x + 1 x ≥ 0x + 1 x < 0

⇒ f(x) is differentiable everywhere.

8. The function f : R – {0} → R given by

( )2x

1 2f x

x e 1= −

−can be made continuous at x = 0 by defining f(0) as(1) 1 (2) 2 (3) – 1 (4) 0

Sol. (1)

2xx 0

1 2lim

x e 1→

−

−

( )

2x

2xx 0

e 1 2x 0lim

0x e 1→

− − = −

2x

2x 2xx 0

2e 2 0lim

0e 1 x.2e→

− = − +

2x

2x 2x 2xx 0

4elim

2e 2e 4xe→

=

+ +4

14

= =

∴ f(0) = 1

9. The solution for x of the equation

x

22

dt

2t t 1

π=

−∫ is

(1) 2 2 (2) 2 (3) π (4)3

2

Sol. Wrong Question

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10.dx

cosx 3 sinx+∫ equals

(1)x

log tan C2 12

π − +

(2)1 x

log tan C2 2 12

π + +

(3)1 x

log tan C2 2 12

π − +

(4)x

log tan C2 12

π + +

Sol. (2)

1 dxI

2 1 3

cosx sinx2 2

=

+

∫

1 dx

2sin xcos cos xsin

6 6

=π π

+∫

1 dx

2sin x

6

=π +

∫

1

cosec x dx2 6

π = + ∫

1 xlogtan C

2 2 12

π = + +

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11. The area enclosed between the curves 2y x and y | x |= = is

(1) 13 (2) 23 (3) 1 (4) 16

Sol. (4)

X

Y

O

(1,1)

y = x

y = x2

Y'

X '(0, 0)

y = – x

Area = ( )1

0

x x dx−∫

1

3 / 2 2

0

x x3 2

2

= −

2 1 1

3 2 6= − = sq. units

12. If the difference between the roots of the equation 2x ax 1 0+ + = is less than 5 , then the set

of possible values of a is

(1) ( ) – , 3∞ − (2) ( )3, 3− (3) ( )3,− ∞ (4) ( )3, ∞

Sol.

5α − β <3

Again a, 1α + β = − αβ =

( )2

5⇒ α −β <

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( )2

4 5⇒ α + β − αβ <

2

a 4 5⇒ − <( )2a 9 a 3, 3⇒ < ⇒ ∈ − ... (i)

Also D ≥ 0a2 – 4 ≥ 0

⇒ ( ) ( )a , 2 2,∈ −∞ − ∪ ∞ ... (ii)

From (i) and (ii)

( ) ( )a 3, 2 2, 3∈ − − ∪

13. In a geometric progression consisting of positive terms, each term equals the sum of the next twoterms. Then the common ratio of this progression equals

(1) ( )1

5 12

− (2) ( )1

1 52

− (3)1

52

(4) 5

Sol. (1)

Let the GP be a, ar, ar2, ar3...∴ a = ar + ar2

⇒ 1 = r + r2

⇒ r2 + r – 1 = 0

1 1 4r2

− ± +∴ =

5 1r

2

−∴ = (3 GP has positive terms)

14. If1 1x 5

sin cosec5 4 2

− − π + =

then a value of x is

(1) 5 (2) 1 (3) 3 (4) 4

Sol. (3)

1 1x 4sin sin

5 5 2

− − π + =

1 1x 3sin cos

5 5 2

− − π ⇒ + =

x 3x 3

5 5⇒ = ⇒ =

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15. In the binomial expansion of ( )n

a b , n 5,− ≥ the sum of 5th and 6th term is zero, thena

bequals

(1)n 4

5

−(2)

5

n 4−(3)

6

n 5−(4)

n 5

6

−

Sol. (1)

5 6T T 0+ =

n n 4 4 n n 5 54 5C a .b C a b 0− −− =

nn 4 45

n 4 5 n

4

Ca b

a b C

−

−⇒ =

( )

( )4! n 4 !a n! n 4

b 5! n 5 ! n! 5

− −⇒ = × =

−

16. The set S : = {1, 2, 3..., 12} is to be partitioned into three sets A, B and C of equal size. Thus,

A B C S, A C B C A C∪ ∪ = ∩ = ∩ = ∩ = φ . The numbers of ways to partition S is

(1)( )

4

12!

3!(2)

( )3

12!

3! 4!(3)

( )4

12!

3! 3!(4)

( )3

12!

4!

Sol. (4)

Total number of ways =( ) ( )

3 3

12! 12!3!

4! 3! 4!× =

×

17. The largest interval lying in ,2 2

−π π

for which the function

( ) ( )

2x 1 x

f x 4 cos 1 log cos x2

− − = + − +

is defined, is

(1) 0,2

π

(2) [ ]0, π (3) ,2 2

π π −

(4) ,4 2

π π − Sol. (1)

( )2x 1 x

f(x) 4 cos 1 log cosx2

− − = + − +

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For2x4 x R− ⇒ ∈ ... (i)

For 1 xcos 12

− −

x x1 1 1 0 2

2 2− ≤ − ≤ ⇒ ≤ ≤

0 x 4⇒ ≤ ≤ ... (ii)

For log(cosx)

cosx > 0 x2 2

−π π⇒ < < ... (iii)

From (i), (ii) and (iii)

x 0,2

π ∈

18. A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends beingfastened to the extremities of a rod 13 m long. If the rod be so held that the body hands immediatelybelow the middle point. The tension in the strings are(1) 5 kg and 13 kg (2) 12 kg and 13 kg (3) 5 kg and 5 kg (4) 5 kg and 12 kg

Sol. (4)

A

G

B

C

13 kg

90°– θ

θ

θ

9 0 ° –

θ 5m

T1

T 2

13 m

12m

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2 2 2 2 2 213 5 12 AB AC BC ACB 90= + ⇒ = + ⇒ ∠ = °

3 G is mid-point of hypotenuse AB.

∴ GA = GB = GC ⇒ GC = 6.5mLet ∠GBC = θ, then, ∠GCB = θBy Lami’s theorem

( ) ( )1 2T T 13

sin 180 sin 90 sin90= =

° − θ ° + θ °

⇒ 1 2T T 13

sin cos 1= =

θ θ

1 2T 13sin and T 13cos⇒ = θ = θ

1 25 12T 13 and T 13

13 13⇒ = × = × as 5 12

sin , cos13 13

θ = θ =

1 2T 5 kg and T 12 kg⇒ = =

19. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly9 twice is

(1)8

243(2)

1

729(3)

8

9(4)

8

729

Sol. (1)

Probability of getting exactly 9 is1

9

and probability of not getting 9 is1 8

19 9

− =

∴ Required probability =

23

21 8

C9 9

×

3! 1 8

2! 81 9= × ×

6 8 8

2 81 9 243

×= =

× ×

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20. Consider a family of circles which are passing through the point ( – 1, 1) and are tangent to x-axis.If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by theinterval

(1)1

k2

≤ (2) 0 < k <1

2(3)

1k

2≥ (4)

1 1k

2 2− ≤ ≤

Sol. (3)

( – 1, 1)C(h, k)

k

( ) ( )2 2 2h 1 k 1 k+ + − =

2 2 2h 2h 1 k 2k 1 k⇒ + + + − + =

( )2h 2h 2 2k 0⇒ + + − =

∴ D ≥ 0⇒ 4 – 4(2 – 2k) ≥ 0⇒ 4 – 8 + 8k ≥ 0⇒ 8k ≥ 4

k ≥ 1

2

21. Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes anangle α with the positive x-axis, then cos α equals.

(a)1

2 (2)1

3 (3)1

2 (4) 1

Sol. (2)

Given planes are

2x 3y z 1 0+ + − = ... (i)

x 3y 2x 2 0+ + − = ... (ii)

Let l, m, n be the direction cosines of line of intersection of plane (i) and (ii).

2l 3m n 0+ + = ... (iii)

l 3m 2n 0+ + = ... (iv)

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Solving (iii) and (iv), we getm = – l, n = l

2 2 2

l m n 1+ + =2 1

3l 1 l3

⇒ = ⇒ = ±

22. The differential equation of all circles passing through the origin and having their centres on thex-axis is

(1) 2 2 dyy x – 2xy

dx= (2) 2 2 dy

x y xydx

= +

(3)

2 2 dy

x y 3xy dx= + (4)

2 2 dy

y x 2xy dx= +Sol. (4)

General equation of circle

2 2x y 2gx 2fy c 0+ + + + =As centre is on x-axis, f = 0As circle is passing through origin, c = 0Equation of required circle will be

2 2x y 2gx 0+ + = ... (i)

Differentiating w.r.t. x, we get

dy2x 2y 2g 0

dx+ + = ... (ii)

Eliminating g from (i) and (ii)

2 2 dyx y x 2x 2y 0

dx

+ + − − =

2 2 dyy x 2xy

dx= +

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23. If p and q are positive real numbers such that p2 + q2 = 1, then the maximum value of (p + q) is

(1)2

(2) 2 (3)1

2(4)

1

2

Sol. (1)

Given 2 2p q 1+ = ... (i)

From (i), we can say 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1∴ Put p = sinθ q = cosθ∴ p + q = sinθ + cosθ

Maximum value of sinθ + cosθ = 2

∴ Maximum value of p + q = 2

24. A tower stands at the centre of a circular park. A and B are two points on the boundary of the parksuch that AB (= a) subtends an angle of 60°at the foot of the tower, and the angle of elevation ofthe top of the tower from A or B is 30°. The height of the tower is

(1) a 3 (2)2a

3(3) 2a 3 (4)

a

3

Sol. (4)

O

P

A

B

30°

60°

30°

h

a

OA = OB = radiiIn ∆OAB, OA = OB = AB = aIn ∆POB

tan30°=h

a

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1 h

a3⇒ =

ah

3⇒ =

25. The sum of the series20C0 – 20C1 + 20C2 – 20C3 + … – … + 20C10 is

(1) 20C10 (2) – 20C10 (3)20

101

C2

(4) 0

Sol. (3)

Given series isx = 20C0 – 20C1 + 20C2 – 20C3 + … – … + 20C10

⇒ 2x = 2 20C0 – 2 20C1 + 2 20C2 + … – … + 2 20C10

( ) ( ) ( ) ( )20 20 20 20 20 20 20 200 20 1 19 2 18 10 10C C C C C C ... C C= + − + + + + + +

( )20 20 20 20 20 20 20 200 1 2 10 18 19 20 102x C C C ... C ... C C C C⇒ = − + + + + + − + +

As 20 20 20 200 1 2 20C C C ... C 0− + + + =

20 20

10 10

1

2x C x C2∴ = ⇒ =

26. The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twicethe abscissa of P, then the curve is a(1) hyperbola (2) ellipse (3) parabola (4) circle

Sol. (1, 2)

Let y = f(x) be a curve

dy

dx∴ = slope of tangent

⇒ dx

dy− = slope of normal

Equation of normal

Y – y = ( )dx

X xdx

− −

dyG x y , 0

dx

∴ ≡ +

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Givendy

x y 2xdx

+ =

dyy x ydy xdx

dx⇒ = ⇒ = or

dyy 3x

dx= = −

2 2y xK or ydy 3xdx

2 2⇒ = + = −

2 2 2 22 2

1y 3x x y

y x K or K or K2 2 2 / 3 2

−⇒ − = = − + + =

∴ Curve is hyperbola or ellipse.

27. If z 4 3,+ ≤ then the maximum value of z 1+ is

(1) 0 (2) 4 (3) 10 (4) 6

Sol. (4)

Given z 4 3+ ≤

( )z 1 z 4 3 z 4 3+ = + + − ≤ + + −

z 1 z 4 3⇒ + ≤ + +

z 1 3 3⇒ + ≤ +z 1 6⇒ + ≤

Maximum value of z 1+ is 6.

28. The resultant of two forces P N and 3 N is a force of 7 N. If the direction of the 3 N force were

reversed, the resultant would be 19 N . The value of P is

(1) 4 N (2) 5 N (3) 6 N (4) 3 N

Sol. (2)

θ

F2

F 1

1F PN=KKH

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2F 3N=KKH

2 21R 3 P 6Pcos 7= + + θ =240 P

cos6P

−θ = ... (i)

θ

F2

F1π−θ – F2

( )22R 9 P 6Pcos 19= + + π − θ =

2P 6Pcos 10− θ =

( )2 2P 40 P 10− − = [(from i)]

22P 50=∴ P = 5N

29. Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hitcorrectly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses thetarget. The probability that the target is hit by the second plane is(1) 0.7 (2) 0.06 (3) 0.14 (4) 0.2

Sol. (3)

Let A is the event of the plane I hit the target correctly.B is the event of the plane II hit the target correctly.

P(A) = .3 ( )cP A .7=

P(B) = .2

( )

cP B .8=

Probability that the target is hit by the second plane = ( ) ( )cP A .P B .7 .2 .14= × =

Assume second plane hit the target only one time.

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30. If

1 1 1

D 1 1 x 1

1 1 1 y

= +

+for x 0,≠ y 0,≠ then D is

(1) divisible by y but not x (2) divisible by neither x nor y(3) divisible by both x and y (4) divisible by x but not y

Sol. (3)

1 1 1

D 1 1 x 1

1 1 1 y

= ++

1 1 2 2 2 3C C C , C C C→ − → −

0 0 1

D x x 1 xy

0 y 1 y

= − =− +

∴ D is divisible by both x and y.

31. For the hyperbola

2 2

2 2

x y1,

cos sin− =

α αwhich of the following remains constant when α varies?

(1) Abscissae of foci (2) Eccentricity(3) Directrix (4) Abscissae of vertices

Sol. (1)

2 2

2 2

x y1

cos sin− =

α α

( )2 2 2sin cos e 1α = α −

2 2

cos e 1 ecos 1∴ α = ⇒ θ = ±Here, a = cosα, b = sinαAbscissae of foci = ±ae = ±ecosα = ±1

1e

cos=

α(depends on α)

2ax cos

e= ± = ± α

Abscissae of vertices = ±a = ±cosα

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32. If a line makes an angle of4

πwith the positive directions of each of x-axis and y-axis, then the

angle that the line makes with the positive direction of z-axis is

(1)2

π(2)

6

π(3)

3

π(4)

4

π

Sol. (1)

1l cos

4 2

π= =

1m cos

42

π= =

2 2 2l m n 1∴ + + =

21 1n 1

2 2⇒ + + =

2n 0 n 0⇒ = ⇒ =

cos 02

π∴ γ = ⇒ γ =

33. A value of C for which the conclusion of Mean Value Theorem holds for the function f(x) = loge x onthe interval [1, 3] is

(1) loge3 (2) 2 log

3e (3) e

1log 3

2(4) 3log e

Sol. (2)

Given function ef(x) log x=Mean Value Theorem for [1, 3]

f(3) f(1)f '(c)

3 1

−=

−

e ee

log 3 log 11 1log 3

c 2 2

−= =

3e

2c 2log e

log 3= =

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34. The function ( )1f(x) tan sinx cos x−= + is an increasing function in

(1) ,2 2−π π

(2) ,4 2π π

(3) ,2 4−π π

(4) 0,2π

Sol. (3)

( )1f(x) tan sinx cos x−= +

Let Z = sinx + cosxf(x) = tan – 1(Z)f(x) is increasing only when Z increases

Z 2 sin x4

π = +

From options Z increases only when x2 4

π π− < <

35. Let

5 5

A 0 5

0 0 5

α α = α α

If2A 25,= then |α| equals

(1) 5 (2) 52 (3) 1 (4) 15

Sol. (4)

5 5

A 0 5

0 0 5

α α = α α

22A 25 A 25= ⇒ =

A 5∴ = ±5 5

A 0 5

0 0 5

α α = α α

= ( )5 5 0 25α − = α

125 5

5∴ α = ± ⇒ α = ±

1

5∴ α =

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36. The sum of the series1 1 1

...2! 3! 4!

− + − up to infinity is

(1)

1

2e+

(2) 2e− (3) 1e− (4)

1

2e−

Sol. (3)

1 1 1...

2! 3! 4!− + − ∞

2 3x x x x

e 1 ...1! 2! 3!

= + + + + ∞

1 1 1 1 1e 1 ...

1! 2! 3! 4!

− = − + − + ∞

11 1 1... e

2! 3! 4!

−− + + ∞ =

37. If ˆ ˆu and v are unit vectors and θ is the acute angle between them, then ˆ ˆ2u 3v× is a unit vector for

(1) Exactly one value of θ (2) Exactly two values of θ(3) More than two values of θ (4) No value of θ

Sol. (1)

θ

v

u

ˆ ˆ2u 3v×

ˆˆ ˆ6 u v sin n= θ

Where n unit vector perpendicular to u and v

ˆˆ ˆ2u 3v 6sin n= × = θ

16sin 1 sin

6θ = ⇒ θ =

∴ Here is one and only one value of θ between 0°and 90°for which1

sin6

θ =

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38. A particle just clears a wall of height b at a distance a and strikes the ground at a distance c fromthe point projection. The angle of projection is

(1) 1 bctana

− (2) 1 btanac

− (3) 45° (4)( )

1 bctana c a

−−

Sol. (4)

θb

S(a,b)

a T M (c, 0)Lc

u

Let u = velocity of projectionθ = angle of projection

2

2 2

gy x tan x

2u cos= θ −

θAs M is on the trajectory.

2

2 2

g0 c tan c

2u cos= θ −

θ

2 2

gtan c

2u cos⇒ θ =

θ ... (i)

2

2 2

gb a tan a

2u cos= θ −

θ... (ii)

From (i) and (ii), we get

( )

bctan

a c aθ =

−

( )1 bc

tana c a

−∴ θ =−

39. The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys andgirls combined is 50. The percentage of boys in the class is(1) 60 (2) 40 (3) 20 (4) 80

Sol. (4)

Let x = Number of boys in the classy = Number of girls in the classSum of marks of all boys = 52x

8/9/2019 Aieee 2007 Math


Code : N

Sum of marks of all girls = 42yAverage of boys and girls combined marks = 50

52x 42y50x y

+= +

⇒ x = 4y

Percentage of boys in the class =x 4y

100 100 80%x y 5y

× = × =+

40. The equation of a tangent to the parabola 2y 8x is y x 2.= = + The point on this line from which

the other tangent to the parabola is perpendicular to the given tangent is(1) ( – 2, 0) (2) ( – 1, 1) (3) (0, 2) (4) (2, 4)

Sol. (1)

Given parabola is y2 = 8x

Given tangent y = x + 2 ...(i)As second tangent is perpendicular to (i), so that pair is on the directrix as directrix is thedirector circle.Equation of direcrix x = – 2 ...(ii)Solving (i) and (ii), we get ( – 2, 0)

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Aieee 2007 Math

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